Failure of tangent classification in positive characteristic

Statement

Over a field $k$ of characteristic $p>0$ , the tangent Lie algebra functor from finite-dimensional formal groups is not an equivalence. Distinct formal groups can have isomorphic tangent Lie algebras because the tangent space records only first-order multiplication, while Frobenius and the $p$ -series begin at higher order.

Basic counterexample

For the additive and multiplicative one-dimensional laws,

F_a(X,Y)=X+Y,\qquad F_m(X,Y)=X+Y+XY,

both tangent Lie algebras are the one-dimensional abelian algebra. Yet

[p]_{F_a}(X)=0, \qquad [p]_{F_m}(X)=(1+X)^p-1=X^p.

An isomorphism of formal group laws must intertwine the $p$ -series, so these two laws are not isomorphic.

Equivalently, the two laws have different heights: $F_a$ has height $\infty$ , while $F_m$ has height $1$ . Height is invisible in the ordinary one-dimensional tangent Lie algebra.

Additional structure is not a complete repair

For suitable group schemes, tangent Lie algebras in characteristic $p$ carry a restricted $p$ -operation. This is an important enrichment, but it still does not replace the full formal group, its Frobenius and Verschiebung operators, or its complete $p$ -series. Cartier–Dieudonné modules and related theories provide finer classifications under additional commutativity, finiteness, and perfection hypotheses.

Consequence for the characteristic-zero theorem

The rational coefficients in the BCH series and formal logarithm are not cosmetic. Division by integers is precisely what allows higher formal data to be reconstructed from the bracket in characteristic zero. No such reconstruction can be asserted after reducing modulo $p$ .

References

Michiel Hazewinkel, Formal Groups and Applications, AMS Chelsea Publishing, 2012. AMS book record. Relevant: Chapters 3–5, $p$ -typical laws, height, and Cartier–Dieudonné theory.
A. Fröhlich, Formal Groups, Lecture Notes in Mathematics 74, Springer, 1968. Publisher record. Relevant: Chapters 3–4, one-dimensional commutative formal groups.